Special Cases of First Order Systems

Pure gain system

\tau=0

No dynamics / very fast dynamics

Examples of pure gain systems

Extremely fast sensors

Fast valves

Pure gain system transfer function

g(s)=K_{p}

For a pure gain system, responses are

Instantaneous scaling

System multiplies input- no lag

Pure gain system: step change equation

x(t)=K_{p}\cdot u(t)

Pure gain system: unit impulse equation

x(t)=K_{p}\cdot\delta(t)

Pure gain system: frequency response

A(\omega)=K_{p}

\phi=0

Integrating system

\frac{1}{\tau}=0

Pole at s = 0

Example of integrating system

Liquid tank with constant outflow and varying inflow

Mass balance for integrating systems

Traditional mass balance

A\rho\cdot\frac{dh}{dt}=F_{IN}-F_{OUT}

Integrating system: deviation form of mass balance

\frac{d\Delta h}{dt}=\frac{1}{A\rho}\cdot\Delta F_{IN}(t)

Found using linearisation method

Integrating system generic transfer function

g(s)=\frac{K_{p}^{\prime}}{s}

K_{p}^{\prime}=\frac{1}{A\rho}

Level integrates flow deviation

Integrating system: step response

Input step of size \Delta F

\Delta h(t)=K_{p}^{\prime}\cdot\Delta Ft

Constant slope

Integrating system: impulse response

\Delta h=K_{p}^{\prime}\cdot V

V is volume of impulse, instant jump to a constant level

Integrating system: frequency response